How can I fix this SubSurf Modifier issue?
This looks like the sort of thing that is best modeled using hard edge surface modelling, with manual placement of edges instead of using a subdivision surface modifier that will round everything off, then keep constantly fighting against it to try and keep sharp corners and tight edges.

Anyway, edge creases seem to work, all you need is an extra inset, then set edge crease to 1 on all corner diagonal edges.



Or alternatively you could instead use full control-edge-loops all around the sphere to keep smoothing from propagating too far.

Select the four edgeloops limiting the indented area and bevel them with the bevel tool Ctrl + B.



Select the relevant parts and extrude them inwards by vertex normal



Now add control edge loops with Ctrl + R around every sharp edge so smoothing won't propagate too far



Repeat for all necessary corners inside and outside where the edges must be sharp